It occurs to me that the princess isn't very bright. Her method of confirming that FutureLucy is telling the truth is seeing if her prediction about the tournament comes true. But that doesn't prove anything. FutureLucy might have psychic powers (do they have those in Fiore?), or might simply have access to information nobody else does. And even if she is from the future, she could still be lying about her true intentions. Of course, I doubt any of that is the case given what manga this is, but the princess really comes off quite gullible.